Now we shall briefly mention the key steps of the proof of Theorem REF  which are similar to the ones of [1]}, [2]}, [3]}. The core of the proof of Theorem REF  relies on Berti-Bolle theory [4]} and [1]}. We mention here some results which have been obtained using this theory [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}. Plugging the ansatz (REF ) into the vortex patch equation (REF ) provides a nonlinear Hamiltonian transport PDE for the radial deformation \(r\)  in the form
\(\partial _{t}r=\partial _{\theta }\nabla {H}(r),\) 
